Problem: If the parabola defined by $y = ax^2 + 6$ is tangent to the line $y = x,$ then calculate the constant $a.$
The parabola $y = ax^2 + 6$ is tangent to the line $y = x$ when the equation
\[ax^2 + 6 = x\]has a double root (which is the $x$-coordinate of the point of tangency).  From this equation,
\[ax^2 - x + 6 = 0.\]This quadratic has a double root when the discriminant is 0, which gives us $1 - 24a = 0.$  Hence, $a = \boxed{\frac{1}{24}}.$